Lemma 28.22.10. Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let \mathcal{A} be a quasi-coherent \mathcal{O}_ X-algebra. There exist
a directed set I (see Categories, Definition 4.21.1),
a system (\mathcal{A}_ i, \varphi _{ii'}) over I in the category of \mathcal{O}_ X-algebras,
morphisms of \mathcal{O}_ X-algebras \varphi _ i : \mathcal{A}_ i \to \mathcal{A}
such that each \mathcal{A}_ i is a quasi-coherent \mathcal{O}_ X-algebra of finite presentation and such that the morphisms \varphi _ i induce an isomorphism
\mathop{\mathrm{colim}}\nolimits _ i \mathcal{A}_ i = \mathcal{A}.
Proof.
First we write \mathcal{A} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 28.22.7. For each i let \mathcal{B}_ i = \text{Sym}(\mathcal{F}_ i) be the symmetric algebra on \mathcal{F}_ i over \mathcal{O}_ X. Write \mathcal{I}_ i = \mathop{\mathrm{Ker}}(\mathcal{B}_ i \to \mathcal{A}). Write \mathcal{I}_ i = \mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}_{i, j} where \mathcal{F}_{i, j} is a finite type quasi-coherent submodule of \mathcal{I}_ i, see Lemma 28.22.3. Set \mathcal{I}_{i, j} \subset \mathcal{I}_ i equal to the \mathcal{B}_ i-ideal generated by \mathcal{F}_{i, j}. Set \mathcal{A}_{i, j} = \mathcal{B}_ i/\mathcal{I}_{i, j}. Then \mathcal{A}_{i, j} is a quasi-coherent finitely presented \mathcal{O}_ X-algebra. Define (i, j) \leq (i', j') if i \leq i' and the map \mathcal{B}_ i \to \mathcal{B}_{i'} maps the ideal \mathcal{I}_{i, j} into the ideal \mathcal{I}_{i', j'}. Then it is clear that \mathcal{A} = \mathop{\mathrm{colim}}\nolimits _{i, j} \mathcal{A}_{i, j}.
\square
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